The reason for considering terms not represented as a sum of three squares is that the description turns out to be more concise (similarly as for non-negative integers). We focus primarily on the case \(m=2^k-1\)  for \(k \in \mathbb {N}_+\)  due to the fact that the 2-adic valuation \(\nu _2(b_{2^k-1}(n))\)  is known to be bounded [1]}, [2]}. This allows us to determine which terms \(b_{2^k-1}(n)\)  are of the form \(4^{r}(8s+7)\)  through the reduction modulo a suitable power of 2.
The analysis is divided into three cases: \(k=1, k=2\) , and \(k \ge 3\) , covered in Section , , and , respectively. Based on these results, in Section  we give quite precise bounds for the counting function of the set \(S_{2^k-1}\) , and determine its natural density in the process. In the final section, we state some questions, problems, and conjectures which may serve as a basis for further study. In particular, we discuss the equation \(b_m(n) = x^2+y^2+z^2\)  when \(m \ne 2^k-1\)  as well as some interesting findings concerning the behavior of \(b_m(n)\)  modulo powers of 2. We also collect results of numerical computations related to the equations \(b(n)=x^2+y^2+z^4\)  and \(b(n)=x^2+y^2\) .
